Numerical approach to an age-structured Lotka-Volterra model

Jordi Ripoll; Jordi Font; Jordi Ripoll; Jordi Font

doi:10.3934/mbe.2023696

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 15603-15622. doi: 10.3934/mbe.2023696

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Numerical approach to an age-structured Lotka-Volterra model

Jordi Ripoll ^,,
Jordi Font

Department of Computer Science, Applied Mathematics and Statistics, University of Girona, Campus Montilivi, 17003 Girona, Spain

Received: 13 June 2023 Revised: 06 July 2023 Accepted: 11 July 2023 Published: 27 July 2023

We study the impact of an age-dependent interaction in a structured predator-prey model. We present two approaches, the PDE (partial differential equation) and the renewal equation, highlighting the advantages of each one. We develop efficient numerical methods to compute the (un)stability of steady-states and the time-evolution of the interacting populations, in the form of oscillating orbits in the plane of prey birth-rate and predator population size. The asymptotic behavior when species interaction does not depend on age is completely determined through the age-profile and a predator-prey limit system of ODEs (ordinary differential equations). The appearance of a Hopf bifurcation is shown for a biologically meaningful age-dependent interaction, where the system transitions from a stable coexistence equilibrium to a collection of periodic orbits around it, and eventually to a stable limit cycle (isolated periodic orbit). Several explicit analytical solutions are used to test the accuracy of the proposed computational methods.
- Structured ecological models,
- chronological age,
- predator-prey models,
- renewal equations,
- efficient numerical methods,
- Hopf bifurcation
Citation: Jordi Ripoll, Jordi Font. Numerical approach to an age-structured Lotka-Volterra model[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 15603-15622. doi: 10.3934/mbe.2023696

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Abstract

We study the impact of an age-dependent interaction in a structured predator-prey model. We present two approaches, the PDE (partial differential equation) and the renewal equation, highlighting the advantages of each one. We develop efficient numerical methods to compute the (un)stability of steady-states and the time-evolution of the interacting populations, in the form of oscillating orbits in the plane of prey birth-rate and predator population size. The asymptotic behavior when species interaction does not depend on age is completely determined through the age-profile and a predator-prey limit system of ODEs (ordinary differential equations). The appearance of a Hopf bifurcation is shown for a biologically meaningful age-dependent interaction, where the system transitions from a stable coexistence equilibrium to a collection of periodic orbits around it, and eventually to a stable limit cycle (isolated periodic orbit). Several explicit analytical solutions are used to test the accuracy of the proposed computational methods.

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